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A novel wavelet based approach for time series data analysis [Elektronische Ressource] / von Thomas Meinl

197 pages
A Novel Wavelet Based Approach forTime Series Data AnalysisZur Erlangung des akademischen Grades einesDoktors der Wirtschaftswissenschaften(Dr. rer. pol.)von der Fakulta¨t fu¨rWirtschaftswissenschaftendes Karlsruhe Instituts fu¨r Technologie (KIT)genehmigteDissertationvonDipl.-Math. Thomas MeinlTag der mu¨ndlichen Pru¨fung: 10.05.2011Referent: Prof. Dr. Svetlozar RachevKorreferent: Prof. Dr. Karl-Heinz Waldmann2011 KarlsruheTo Jack.AcknowledgementsThis work would not have been possiblewithout the loving supportof all my family, mydog, and my ex-wife. Thank you all, I am most deeply obliged to you for being alongmy side over all these years.IalsothankmysupervisorProf.Dr.SvetlozarRachevandDr.EdwardSunfortheirguid-anceandtheir invaluableinputwhichwithoutthisthesis couldnothave beenfinishedinsuch a short time and with such excellent results. I also thank Christof Weinhardt whoprovided me with the opportunity to undertake some serious research at his institute,nottomention thefunIhadwithall thenicecolleagues. Thankyou, wehadsometimestogether we will remember for sure.During the time this thesis was written the author undertook a stay abroad in Brazil aswell as in Japan, which was only madefeasible thanks to thehighly appreciated supportof the Karlsruhe House of Young Scientists. My thanks also goes to all the people therewho welcomed me so heartily.
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A Novel Wavelet Based Approach for
Time Series Data Analysis
Zur Erlangung des akademischen Grades eines
Doktors der Wirtschaftswissenschaften
(Dr. rer. pol.)
von der Fakulta¨t fu¨r
Wirtschaftswissenschaften
des Karlsruhe Instituts fu¨r Technologie (KIT)
genehmigte
Dissertation
von
Dipl.-Math. Thomas Meinl
Tag der mu¨ndlichen Pru¨fung: 10.05.2011
Referent: Prof. Dr. Svetlozar Rachev
Korreferent: Prof. Dr. Karl-Heinz Waldmann
2011 KarlsruheTo Jack.Acknowledgements
This work would not have been possiblewithout the loving supportof all my family, my
dog, and my ex-wife. Thank you all, I am most deeply obliged to you for being along
my side over all these years.
IalsothankmysupervisorProf.Dr.SvetlozarRachevandDr.EdwardSunfortheirguid-
anceandtheir invaluableinputwhichwithoutthisthesis couldnothave beenfinishedin
such a short time and with such excellent results. I also thank Christof Weinhardt who
provided me with the opportunity to undertake some serious research at his institute,
nottomention thefunIhadwithall thenicecolleagues. Thankyou, wehadsometimes
together we will remember for sure.
During the time this thesis was written the author undertook a stay abroad in Brazil as
well as in Japan, which was only madefeasible thanks to thehighly appreciated support
of the Karlsruhe House of Young Scientists. My thanks also goes to all the people there
who welcomed me so heartily.
Karlsruhe, May 2011
Thomas MeinlAbstract
Time series analysis is still a very wide field of research from both a theoretical point of
view as well as amongst practitioners. Among the very first tasks in the analysis proce-
dureis theestimation oflong-term trends, thatis, theseparation ofthisgenerally slowly
evolvingcomponentfromanyshort-termfluctuations. Usually, thetrendcurve,whichin
most cases is expected to besmooth, can be extracted by a variety of different methods.
However, in many application scenarios thetrend mustalso account for suddenchanges.
These sudden changes comprise of not only jumps, but also other phenomena like steep
slopes and valleys. This challenge constitutes an on-going problem for traditional trend
estimation methods. While established filtering techniques either fail to capture these
sudden changes accurately or are sensitive to high-amplitude fluctuations, the applica-
tion of parametric methods is challenging due to the generally unknown trend and the
innumerable shapes that these sudden changes can assume.
This thesis proposesa trend extraction approach based on wavelet methods. Thenew
algorithm, named local linear scaling approximation (LLSA), is developed by analyzing
specific wavelet coefficient step response structures and by transferring these structures
onto real signals. This procedure enables the analyst to extract a trend whose smooth-
ness is comparable to the output of linear filtering techniques, while at the same time
capturing the details of sudden changes with arbitrary shapes, an area in which usu-
ally most nonlinear filters excel. Therefore, LLSA can be seen as a novel approach to
bridge the gap between linear and nonlinear filters. The algorithm was developed to be
applicable on homogeneous time series without any further requirements on these, and
to work with only two additional input parameters, which can also be set in a heuristic
manner, yielding a directly implementable and usable method.
vMoreover, thealgorithm’s propertiesareshown, namelyitscomputational complexity,
its local linearity, and its impulse and step response. The robustness of LLSA is first
shown analytically, and then substantiated by several analyses performed on simulated
signals as well as on empirical data. LLSA’s performance is further evaluated in two
separate application scenarios, that are, price volatility estimation and value at risk.
The algorithm’s superior performance in relation to two benchmark filtering techniques
is shown for a considerable number of cases, and several aspects (i.e., possibilities and
limitations) of LLSA’s general application are discussed.
viContents
List of Figures xi
List of Tables xiii
1. Introduction 1
1.1. Requirements and Research Questions . . . . . . . . . . . . . . . . . . . . 3
1.2. Contributions of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3. Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Methods of Trend Extraction 9
2.1. Time Series Analysis and Trend Extraction . . . . . . . . . . . . . . . . . 9
2.2. Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1. General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2. Transfer Functions: Time vs. Frequency Domain . . . . . . . . . . 23
2.3. Nonlinear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1. General Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2. Filter Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4. Further Related Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1. Algorithms for Jump Detection and Modeling . . . . . . . . . . . . 35
2.4.2. Alternative Methods for Trend Estimation . . . . . . . . . . . . . . 37
2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3. Wavelets and Their Transforms 45
3.1. Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2. Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
viiContents
3.3. Wavelet Trend Extraction and Denoising Methods . . . . . . . . . . . . . 62
3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4. The Local Linear Scaling Approximation 67
4.1. Methodology and Implementation. . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1. Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.2. Final Formulation and Remarks . . . . . . . . . . . . . . . . . . . 75
4.1.3. Implementation and Usage . . . . . . . . . . . . . . . . . . . . . . 78
4.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1. Computational Complexity . . . . . . . . . . . . . . . . . . . . . . 81
4.2.2. Local Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.3. Impulse and Step Response . . . . . . . . . . . . . . . . . . . . . . 83
4.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5. Evaluation and Application 87
5.1. Robustness and Performance Studies . . . . . . . . . . . . . . . . . . . . . 87
5.1.1. Analytical Consistency . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.2. Simulated Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.3. Empirical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.1. General Application and Examples . . . . . . . . . . . . . . . . . . 102
5.2.2. Price Volatility Estimation . . . . . . . . . . . . . . . . . . . . . . 110
5.2.3. Estimating Value at Risk of High-Frequency Data . . . . . . . . . 114
5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6. Conclusion and Outlook 119
6.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1.1. Requirement Satisfaction and Research Questions . . . . . . . . . 119
6.1.2. Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2. Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.1. Algorithmic Extensions . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.2. Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 129
viii